Many moons ago I used to do the proof reading for publicity leaflets like the rovers and this quirk raised it's head then. If I remember the answer I got then was that, although the rover is valid on the various operators in the area, as the FOSW▸ (Freedom of (the) South West Rover) was a GWR▸ (Great Western Railway) sponsored item it would only be valid on lines where GWR actually ran a service, although their argument fell down when it came to the SWR» (South Western Railway - about) line from Axminster to Salisbury!!
It did seem a bit petty to me at the time but, brick walls and head banging.....
As an aside, if you down load the current map from the GWR website you'll see that both the Chandlers Ford line and Lymington Pier branch are not there..
https://www.gwr.com/stations-and-destinations/stations